Question 48

We are given the 4-vertex distance matrix:

The edge with variable weight $x$ is $e = \{3, 4\}$.

For edge $\{3, 4\}$ to be contained in at least one shortest path between some pair of vertices, it must at least be a shortest path between its own endpoints 3 and 4 (or contribute to another shortest path).

Find all alternate paths between vertex 3 and vertex 4 without using edge $\{3, 4\}$:

1. Path $3 - 1 - 4$: weight $= w(3, 1) + w(1, 4) = 8 + 5 = 13$
2. Path $3 - 2 - 4$: weight $= w(3, 2) + w(2, 4) = 5 + 8 = 13$
3. Path $3 - 2 - 1 - 4$: weight $= w(3, 2) + w(2, 1) + w(1, 4) = 5 + 2 + 5 = 12$

The minimum weight among all alternate paths connecting 3 and 4 is 12.
If $x \le 12$, the direct path $3 - 4$ has weight $x \le 12$, which is less than or equal to the alternate path weight 12, making it a valid shortest path between 3 and 4.
If $x > 12$, the path $3 - 2 - 1 - 4$ has weight 12, strictly shorter than direct edge $x$, and edge $\{3, 4\}$ cannot be part of any shortest path between any pair of vertices.

Therefore, the largest integer value of $x$ is 12.